Existence of Nash equilibrium for Chance-Constrained
Games
Vikas Vikram Singha , Oualid Jouinib , Abdel Lissera
a Laboratoire
de Recherche en Informatique Université Paris Sud, 91405 Orsay, France.
Génie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, 92 290
Châtenay-Malabry, France.
b Laboratoire
Abstract
We consider an n-player strategic game with finite action sets and random
payoffs. We formulate this as a chance-constrained game by considering that the
payoff of each player is defined using a chance-constraint. We consider that the
components of the payoff vector of each player are independent normal/Cauchy
random variables. We also consider the case where the payoff vector of each
player follows a multivariate elliptically symmetric distribution. We show the
existence of a Nash equilibrium in both cases.
Keywords: Chance-constrained game, Elliptically symmetric distribution,
Normal/Cauchy distribution, Nash equilibrium.
1. Introduction
In 1928, John von Neumann [1] showed that there exists a mixed strategy
saddle point equilibrium for a two player zero sum game with finite number of
actions for each player. In 1950, John Nash [2] showed that there always exists
a mixed strategy Nash equilibrium for an n-player general sum game with finite
number of actions for each player. In both [1, 2], it is considered that the players’ payoffs are deterministic. However, there can be practical cases where the
players’ payoffs are better modeled by random variables following certain distributions. The wholesale electricity markets are the good examples that capture
Email addresses: [email protected] (Vikas Vikram Singh), [email protected]
(Oualid Jouini), [email protected] (Abdel Lisser)
Preprint submitted to Elsevier
February 7, 2016
this situation. A recent paper by Mazadi et al. [3] introduced wind integration
on electricity markets due to which the payoffs become random variables. In
some cases the consumers’ demands are random that introduce randomness in
the firms’ payoffs [4, 5, 6]. One way to handle such type of games is to replace
random variables by their expected values and solve the corresponding deterministic game [5, 6]. We hereafter mention few recent papers on the games with
random payoffs using expected payoff criterion. Ravat and Shanbhag [7] studied stochastic Nash games where the payoff function of each player is random
and each player solves expected value problem. They consider both smooth and
non-smooth stochastic Nash games. In various cases they showed the existence
of Nash equilibrium for stochastic Nash games. Xu and Zhang [8], Jadamba and
Raciti [9], considered stochastic Nash equilibrium problems. To solve stochastic
Nash equilibrium problems, a sample average approximation method is used in
[8], and in [9] a variational inequality approach on probabilistic Lebesgue spaces
is used. DeMiguel and Xu [10] proposed a two stage multiple-leader stochastic
Stackelberg Nash-Cournot model where leaders use expected payoff criterion.
The expected payoff criterion does not take a proper account of stochasticity
in the cases where the observed sample payoffs are large amounts with very small
probabilities. To handle such cases the concept of satisficing has been considered
in the literature, where a player is interested in a strategy which maximizes his
total payoff that can be obtained with at least a given probability. Such payoff
criterion is defined using chance-constrained programming [11, 12, 13], and due
to which we call such games chance-constrained games. The papers [3, 4] mentioned above on electricity markets use chance-constrained game formulation to
study the situation. In [3], the randomness in payoffs is due to the installation of
wind generators on the electricity market. The authors consider the case where
the random variables that represent the amount of wind are independent normal random variables, and they also consider the case where the random vector
follows a multivariate normal distribution. Later, for better representation and
ease in computation they discuss, in detail, the case where wind generator is
present at only one bus station. In [4], the consumers’ random demand is as2
sumed to be normally distributed. In [3, 4], the action sets of the players are not
finite. In [3], the game problem is formulated as an equivalent linear complementarity problem (LCP). Hence, the existence of Nash equilibrium depends on
whether the corresponding LCP has a solution. The existence of Nash equilibrium for the games where the action sets are not finite is not easy to show even
when the payoffs are deterministic. It depends on the nature of action sets and
payoff functions [14, 15]. There is also a game theoretic situation in electricity
market where the action sets are finite [16]. Although the players’ payoffs are
deterministic in [16], the counterpart of the model where the payoffs are random
variables using chance-constrained game formulation can be considered. Only
few theoretical results on zero sum chance-constrained games with finite action
sets of the players are available in the literature so far [17, 18, 19, 20].
In this paper, we focus on the games where the payoffs of the players are random variables with known probability distributions. The case where probability
distribution of the random payoffs are not known completely is considered in
[21]. The authors use distributionally robust approach to handle these games.
To the best of our knowledge there is no result on the existence of Nash equilibrium for chance-constrained games even when the action sets of all the players
are finite. We consider an n-player game where the action set of each player is
finite and the payoff vector of each player is a random vector. We formulate
this as a chance-constrained game by considering that the payoff of each player
is defined using a chance-constraint. We first consider the case where the components of the payoff vector of each player are independent random variables.
Specifically, we discuss the case of normal and Cauchy random variables. We
also consider the case where the components of the random payoff vector of each
player are dependent random variables and we assume that the payoff vector of
each player follows a multivariate elliptically symmetric distribution. For each
case we show that there always exists a mixed strategy Nash equilibrium for the
corresponding chance-constrained game.
The structure of the rest of the paper is as follows: in Section 2 we give
the definition of a chance-constrained game. Existence of mixed strategy Nash
3
equilibrium is then given in Section 3.
2. The Model
We consider an n-player strategic game. Let I = {1, 2, · · · , n} be a set of
all players. For each i ∈ I, let Ai be a finite action set of player i and its
generic element is denoted by ai . A vector a = (a1 , a2 , · · · , an ) denotes an
action profile of the game. Let A=×ni=1 Ai be the set of all action profiles of the
game. Denote, A−i =×nj=1;j6=i Ai , and a−i ∈ A−i is a vector of actions aj , j 6= i.
The action set Ai of player i is also called the set of pure strategies of player i.
A mixed strategy of a player is represented by a probability distribution over
his action set. For each i ∈ I, let Xi be the set of mixed strategies of player i,
i.e., it is the set of all probability distributions over the action set Ai . A mixed
strategy τi ∈ Xi is represented by τi = (τi (ai ))ai ∈Ai , where τi (ai ) ≥ 0 is a
P
probability with which player i chooses an action ai and
ai ∈Ai τi (ai ) = 1.
Let X=
×ni=1 Xi
be the set of all mixed strategy profiles of the game and its
element is denoted by τ = (τi )i∈I . Denote, X−i =×nj=1;j6=i Xi , and τ−i ∈ X−i
is a vector of mixed strategies τj , j 6= i. We define (νi , τ−i ) to be a strategy
profile where player i uses strategy νi and each player j, j 6= i, uses strategy
τj . Let ri = (ri (a))a∈A be a payoff vector of player i whose components are real
numbers. Specifically, player i gets payoff ri (a) ∈ R at action profile a. For such
games, Nash [2] showed that there always exists a Nash equilibrium in mixed
strategies.
In real life applications the players’ payoffs may be random due to uncertainty caused by various external factors. Therefore, we consider the case where
the payoffs of each player are random variables and follow a certain distribution. We denote the random payoff vector by riw = (riw (a))a∈A , where w is an
uncertainty parameter. Let (Ω, F, P ) be a probability space. Then, for each
i ∈ I, riw : Ω → R|A| , where |A| is the cardinality of set A. For a given strategy
profile τ ∈ X the payoff of player i, i ∈ I, is defined by
riw (τ ) =
n
XY
a∈A j=1
4
τj (aj )riw (a).
(2.1)
From (2.1), riw (τ ) is a random variable for all τ ∈ X, and it is a linear combination of the components of the random payoff vector riw = (riw (a))a∈A . We
assume that each player uses satisficing payoff criterion, i.e., the payoff function
of each player is defined using a chance-constraint. At strategy profile τ ∈ X,
the payoff of each player is the highest level of his payoff that he can attain with
at least a specified level of confidence. The confidence level of each player is
given a priori and it is known to other players. Let αi ∈ [0, 1] be the confidence
level of player i and α = (αi )i∈I be a confidence level vector. For a given strategy profile τ ∈ X, and a given confidence level vector α the payoff function of
player i, i ∈ I, is given by
w
i
uα
i (τ ) = sup{u|P (ri (τ ) ≥ u) ≥ αi }.
(2.2)
We assume that the probability distributions of the payoffs of each player are
known to all the players. Then, for a given α ∈ [0, 1]n , the payoff function
of a player defined by (2.2) is known to all the players. That is, for a given
α ∈ [0, 1]n the above chance constrained game is a non-cooperative game with
complete information. The set of best response strategies of player i, i ∈ I,
against a given strategy profile τ−i of other players is given by
αi
i
BRiαi (τ−i ) = {τ̄i ∈ Xi |uα
i (τ̄i , τ−i ) ≥ ui (τi , τ−i ), ∀ τi ∈ Xi } .
Next, we give the definition of Nash equilibrium.
Definition 2.1 (Nash equilibrium). A strategy profile τ ∗ ∈ X is said to be
a Nash equilibrium for a given α ∈ [0, 1]n , if for all i ∈ I the following inequality
holds,
αi
∗ ∗
∗
i
uα
i (τi , τ−i ) ≥ ui (τi , τ−i ), ∀ τi ∈ Xi .
∗
That is, τ ∗ is a Nash equilibrium if and only if τi∗ ∈ BRiαi (τ−i
) for all i ∈ I.
3. Existence of Nash equilibrium
We assume that the payoffs of each player are random variables following a
certain distribution. We consider various cases and show the existence of a mixed
strategy Nash equilibrium of chance-constrained game for different values of α.
5
3.1. Payoffs following normal/Cauchy distribution
For a given strategy profile τ ∈ X, the probability distribution of random
payoff riw (τ ), i ∈ I, plays an important role in defining the payoff function of
player i given by (2.2). For each i ∈ I, riw (τ ) is a linear combination of the
components of the random payoff vector riw = (riw (a))a∈A . We consider the
probability distributions that are closed under the linear combination. It is
well known that the independent random variables following normal or Cauchy
distributions possess this property [22]. We first consider the case where for
each i ∈ I, {riw (a)}a∈A are independent normal random variables, where the
mean and variance of riw (a) are µi (a) and σi2 (a) respectively. Then, for τ ∈ X,
P
Qn
riw (τ ) follows a normal distribution with mean µi (τ ) =
a∈A
j=1 τj (aj )µi (a)
P
Qn
riw (τ )−µi (τ )
2
2
2
N
and variance σi (τ ) =
follows
a∈A
j=1 τj (aj )σi (a), and Zi =
σi (τ )
a standard normal distribution. Let FZ−1
N (·), i ∈ I, be a quantile function of a
i
standard normal distribution. From (2.2), for a given τ ∈ X and α, we have
w
i
uα
i (τ ) = sup{u|P (ri (τ ) ≥ u) ≥ αi }
u − µi (τ )
= sup u|P ZiN ≤
≤ 1 − αi
σi (τ )
= µi (τ ) + σi (τ )FZ−1
N (1 − αi ).
i
That is,
i
uα
i (τ ) =
n
XY
a∈A j=1
τj (aj )µi (a) +
n
XY
a∈A j=1
! 21
τj2 (aj )σi2 (a)
FZ−1
N (1 − αi ), i ∈ I.
i
(3.1)
i
Lemma 3.1. uα
i (·, τ−i ), i ∈ I, defined by (3.1) is a concave function of τi for
all αi ∈ [0.5, 1].
i
Proof. For fixed τ−i , the first term of uα
i (·, τ−i ) defined by (3.1) is a linear
i
function of τi . If αi ∈ [0.5, 1], then the second term of uα
i (·, τ−i ) is a concave
P
21
Qn
2
2
is a convex function of τi ,
function of τi because
a∈A
j=1 τj (aj )σi (a)
6
and FZ−1
N (1 − αi ) ≤ 0 for all αi ∈ [0.5, 1]. Hence, we can say that for each i ∈ I,
i
i
uα
i (·, τ−i ) is a concave function of τi for all αi ∈ [0.5, 1].
Theorem 3.2. Consider an n-player game where each player has finite number
of actions. If for each player i, i ∈ I, {riw (a)}a∈A are independent random variables, where riw (a) follows a normal distribution with mean µi (a) and variance
σi2 (a), there exists a mixed strategy Nash equilibrium for all α ∈ [0.5, 1]n .
Proof. Let P(X) be a power set of X. Define a set valued map G : X → P(X)
n
Q
such that G(τ ) =
BRiαi (τ−i ). A strategy profile τ ∈ X is said to be a fixed
i=1
point of the set valued map G if τ ∈ G(τ ). It is easy to see that a fixed point
of G is a Nash equilibrium. Then, it is sufficient to show that G has a fixed
point. In order to show that G has a fixed point, we show that G satisfies all
the following conditions of Kakutani fixed point theorem [23]:
1. X is a non-empty, convex, and compact subset of a finite dimensional
Euclidean space.
2. G(τ ) is non-empty and convex for all τ ∈ X.
3. G(·) has closed graph: If (τn , τ̄n ) → (τ, τ̄ ) with τ̄n ∈ G(τn ) for all n then
τ̄ ∈ G(τ ).
Condition 1 holds from the definition of X. Fix α ∈ [0.5, 1]n . For fixed τ−i ,
αi
i
uα
i (·, τ−i ) is a continuous function of τi from (3.1). For each i ∈ I, BRi (τ−i ),
i
is non-empty because a continuous function uα
i (·, τ−i ) over a compact set Xi
always attains maxima. Hence, G(τ ) is non-empty for all τ ∈ X. For each
i
i ∈ I, BRiαi (τ−i ) is a convex set because uα
i (·, τ−i ) given by (3.1) is a concave
function of τi from Lemma 3.1. Hence, G(τ ) is a convex set for all τ ∈ X.
Now, we prove that G(·) is a closed graph. Assume that G(·) is not a closed
graph, i.e., there is a sequence (τ n , τ̄ n ) → (τ, τ̄ ) with τ̄ n ∈ G(τ n ) for all n, but
τ̄ ∈
/ G(τ ). In this case τ̄i ∈
/ BRi (τ−i ) for some i ∈ I. Then, there is an > 0
and a τ̃i such that
αi
i
uα
i (τ̃i , τ−i ) > ui (τ̄i , τ−i ) + 3.
7
(3.2)
αi
αi n n
i
Since, uα
i (·) is a continuous function of τ from (3.1), ui (τ̄i , τ−i ) → ui (τ̄i , τ−i ).
Then, there exists an integer N1 such that
αi
n n
i
uα
i (τ̄i , τ−i ) < ui (τ̄i , τ−i ) + , ∀ n ≥ N1 .
(3.3)
From (3.2) and (3.3), we have
αi
n n
i
uα
i (τ̄i , τ−i ) < ui (τ̃i , τ−i ) − 2, ∀ n ≥ N1 .
(3.4)
αi
n
i
Similarly, uα
i (τ̃i , τ−i ) → ui (τ̃i , τ−i ). Then, there exists an integer N2 such that
αi
n
i
uα
i (τ̃i , τ−i ) < ui (τ̃i , τ−i ) + , ∀ n ≥ N2 .
(3.5)
Let N = max{N1 , N2 }. Then, from (3.4) and (3.5), we have
αi n n
n
i
uα
i (τ̃i , τ−i ) > ui (τ̄i , τ−i ) + , ∀ n ≥ N.
n
That is, τ̃i performs better than τ̄in against τ−i
for all n ≥ N which contradicts
n
τ̄in ∈ BRiαi (τ−i
) for all n. Hence, G(·) is a closed graph. That is, the set valued
map G(·) satisfies all the conditions of Kakutani fixed point theorem. Hence,
G(·) has a fixed point τ ∗ . Such τ ∗ is a Nash equilibrium of the game. The
confidence level vector α ∈ [0.5, 1]n is arbitrary, therefore, there always exists a
mixed strategy Nash equilibrium for all α ∈ [0.5, 1]n .
Now, we consider the case where for each i ∈ I, {riw (a)}a∈A are independent
Cauchy random variables, where the location and scale parameters of riw (a) are
µi (a) and σi (a) respectively. Then, for a given τ ∈ X, riw (τ ), i ∈ I, follows a
P
Qn
Cauchy distribution with location parameter µi (τ ) = a∈A j=1 τj (aj )µi (a),
P
Qn
and scale parameter σi (τ ) = a∈A j=1 τj (aj )σi (a) because the components
of τ are non-negative [22]. Hence, ZiC =
distribution. Let
FZ−1
C (·)
i
riw (τ )−µi (τ )
σi (τ )
follows a standard Cauchy
be a quantile function of a standard Cauchy distribu-
tion. Similar to the case of normal distribution, for a given τ ∈ X and α we
have from (2.2),
i
uα
i (τ ) =
n
XY
a∈A j=1
τj (aj )µi (a) +
n
XY
a∈A j=1
8
τj (aj )σi (a)FZ−1
C (1 − αi ), i ∈ I.
i
(3.6)
The quantile function FZ−1
C (·) of Cauchy distribution is not finite at 0 and 1.
i
Therefore, we consider the case of αi ∈ (0, 1), i ∈ I, so that the payoff function
i
defined by (3.6) is finite. For fixed τ−i ∈ X−i the function uα
i (·, τ−i ), i ∈ I,
defined by (3.6) is a linear function of τi .
Theorem 3.3. Consider an n-player game where each player has finite number
of actions. If for each player i, i ∈ I, {riw (a)}a∈A are independent random
variables, where riw (a) follows a Cauchy distribution with location parameter
µi (a) and scale parameter σi (a), there exists a mixed strategy Nash equilibrium
for all α ∈ (0, 1)n .
Proof. For each i ∈ I, BRiαi (τ−i ) is a convex set for all αi ∈ (0, 1) because
αi
i
uα
i (·, τ−i ) defined by (3.6) is a linear function of τi . From (3.6), ui (·), i ∈ I, is
a continuous function of τ . Then, the proof follows from the similar arguments
used in Theorem 3.2.
3.1.1. Special cases
In previous section, we show that a Nash equilibrium of chance-constrained
game exists, for all α ∈ [0.5, 1]n , for the case where all the components of
random payoff vector riw are independent normal random variables. We now
consider a special case where the components of the payoff vector of each player
corresponding to only one of his action are independent normal random variables
and rest of the components are deterministic. We show that there exists a Nash
equilibrium for all α ∈ [0, 1]n . Further, if only one component of each player’s
payoff vector is a random variable and all other components are deterministic,
the Nash equilibrium existence for all α ∈ [0, 1]n can be extended to all the
continuous probability distributions whose quantile function exist.
(i) Normal random payoffs of each player corresponding to only one of his action
We assume that for each player i, i ∈ I, there exists an action āi ∈ Ai
such that {riw (āi , a−i )}a−i ∈A−i are independent normal random variables and all
other components of riw are deterministic. That is, for each i ∈ I, riw (ai , a−i ) =
ri (ai , a−i ) for all ai ∈ Ai , ai 6= āi , and a−i ∈ A−i . Then, for each i ∈ I,
9
σi2 (ai , a−i ) = 0, and µi (ai , a−i ) = ri (ai , a−i ), for all ai ∈ Ai , ai 6= āi , and
a−i ∈ A−i . By putting these values in (3.1), we have,
i
uα
i (τ )
X
=
X
τi (ai )
!
τj (aj )ri (ai , a−i )
a−i ∈A−i j=1;j6=i
ai ∈Ai ;ai 6=āi
"
n
Y
X
+ τi (āi )
n
Y
τj (aj )µi (āi , a−i )
a−i ∈A−i j=1;j6=i
n
Y
X
+
! 21
τj2 (aj )σi2 (āi , a−i )
a−i ∈A−i j=1;j6=i
#
FZ−1
N (1
i
− αi ) , i ∈ I. (3.7)
Theorem 3.4. Consider an n-player game where each player has finite number
of actions. If for each player i, i ∈ I, there exists an action āi ∈ Ai such
that {riw (āi , a−i )}a−i ∈A−i are independent random variables, where riw (āi , a−i )
follows a normal distribution with mean µi (āi , a−i ) and variance σi2 (āi , a−i ),
and all other components of riw are deterministic, there exists a mixed strategy
Nash equilibrium for all α ∈ [0, 1]n .
Proof. For each i ∈ I, BRiαi (τ−i ) is a convex set for all αi ∈ [0, 1] because
αi
i
uα
i (·, τ−i ) defined by (3.7) is a linear function of τi . From (3.7), ui (·) is a
continuous function of τ . Then, the proof follows from the similar arguments
used in Theorem 3.2.
(ii) Payoff with one random variable component
We assume that for each player i, i ∈ I, there exists an action profile ai ∈ A
such that riw (ai ) is a normal random variable and all other components of riw
are deterministic, i.e., riw (a) = ri (a) for all a ∈ A, a 6= ai . Then µi (a) = ri (a)
and σi2 (a) = 0 for all a ∈ A, a 6= ai . By putting these values in (3.1), we have,
i
uα
i (τ ) =
X
n
Y
τj (aj )ri (a) +
a∈A;a6=ai j=1
n
Y
τj (aij ) µi (ai ) + σi (ai )FZ−1
.
N (1 − αi )
i
j=1
(3.8)
Let Fr−1
w
i (·) denote the
i (a )
Then, Fr−1
w
i (1 − αi ) =
i (a )
riw (ai ).
quantile function of normal random variable
µi (ai ) + σi (ai )FZ−1
Hence, we can write
N (1 − αi ) .
i
10
(3.8) as
i
uα
i (τ ) =
n
XY
τj (aj )r̃i (a), i ∈ I,
a∈A j=1
where,
r̃i (a) =


i
F −1
r w (ai ) (1 − αi ), if a = a ,
i
(3.9)

ri (a), if a 6= ai .
Hence, the game is equivalent to a deterministic strategic game with payoff
vector r̃i = (r̃i (a))a∈A , i ∈ I, where r̃i (a) ∈ R is defined by (3.9). Therefore,
the existence of Nash equilibrium for all α ∈ [0, 1]n follows from [2]. Since, the
deterministic payoff vector r̃i depends only on the quantile function of normal
random variable riw (ai ), then the result on the existence of Nash equilibrium in
this case can be extended for all the continuous probability distributions whose
quantile function exists.
3.2. Payoffs following multivariate elliptical distributions
In Section 3.1, we consider the case where the components of the random
payoff vector of each player are independent random variables. Here, we consider the case where the components of the random payoff vector of each player
are dependent random variables. We assume that the payoff vector of each
player follows a multivariate elliptically symmetric distribution. The distributions belonging to the class of elliptically symmetric distributions generalize the
multivariate normal distribution. Some famous multivariate distributions like
normal, Cauchy, t, Laplace, and logistic distributions belong to the family of elliptically symmetric distributions. For more details about elliptically symmetric
distributions see [24].
We assume that the payoff vector (riw (a))a∈A of player i, i ∈ I, follows a
multivariate elliptically symmetric distribution with parameters µi and Σi . The
vector µi = (µi (a))a∈A represents a location parameter and Σi is a scale matrix.
We assume Σi to be a positive definite matrix. Then, all linear combinations of
the components of the payoff vector follow a univariate elliptically symmetric
distribution [24]. For a given τ ∈ X, let η τ = (η τ (a))a∈A be a vector, where
11
η τ (a) =
Qn
j=1 τj (aj ).
Then for τ ∈ X, riw (τ ), i ∈ I, follows a univariate ellipti-
cally symmetric distribution with parameters µTi η τ and (η τ )T Σi η τ ; T denotes
p
transposition. Since, Σi is a positive definite matrix, then, (η τ )T Σi η τ will be
a norm and it is denoted by ||η τ ||Σi . For each i ∈ I, ZiS =
τ
riw (τ )−µT
i η
||η τ ||Σi
follows a
univariate spherically symmetric distribution with parameters 0 and 1 [24]. Let
FZ−1
S (·) be a quantile function of a univariate spherically symmetric distribution.
i
From (2.2), for a given τ ∈ X, we obtain
w
i
uα
i (τ ) = sup{u|P (ri (τ ) ≥ u) ≥ αi }
w
u − µTi η τ
ri (τ ) − µTi η τ
= sup u|P
≤
≤ 1 − αi
||η τ ||Σi
||η τ ||Σi
= µTi η τ + ||η τ ||Σi FZ−1
S (1 − αi ).
i
That is,
−1
T τ
τ
i
uα
i (τ ) = µi η + ||η ||Σi FZ S (1 − αi ), i ∈ I.
(3.10)
i
S
We know that the quantile function FZ−1
S (1 − αi ) ≤ 0 for all αi ∈ (0.5, 1]. If Zi
i
has strictly positive density then FZ−1
S (1 − αi ) ≤ 0 for all αi ∈ [0.5, 1] (see [25]).
i
i
Lemma 3.5. uα
i (·, τ−i ), i ∈ I, defined by (3.10) is a concave function of τi for
all αi ∈ (0.5, 1].
Proof. Fix i ∈ I, αi ∈ (0.5, 1], and τ−i ∈ X−i . Let τi1 , τi2 ∈ Xi . Take
λ ∈ [0, 1]. Then, for a strategy profile (λτi1 + (1 − λ)τi2 , τ−i ) we have
Y
1
2
η (λτi +(1−λ)τi ,τ−i ) (a) = λτi1 (ai ) + (1 − λ)τi2 (ai )
τj (aj ),
j∈I;j6=i
1
2
1
2
for all a ∈ A. That is, η (λτi +(1−λ)τi ,τ−i ) = λη (τi ,τ−i ) + (1 − λ)η (τi ,τ−i ) . From
(3.10), we have
1
2
1
2
T
i
uα
λη (τi ,τ−i ) + (1 − λ)η (τi ,τ−i )
i (λτi + (1 − λ)τi , τ−i ) = µi
1
2
+ ||λη (τi ,τ−i ) + (1 − λ)η (τi ,τ−i ) ||Σi FZ−1
S (1 − αi )
i
≥
1
i
λuα
i (τi , τ−i )
+ (1 −
2
i
λ)uα
i (τi , τ−i ).
i
That is, uα
i (·, τ−i ) is a concave function of τi for all αi ∈ (0.5, 1].
12
Remark 3.6. If the random payoff vector (riw (a))a∈A , i ∈ I, has strictly positive density then Lemma 3.5 holds for all αi ∈ [0.5, 1].
Theorem 3.7. Consider an n-player game where each player has finite number
of actions. If the random payoff vector (riw (a))a∈A of player i, i ∈ I, follows
a multivariate elliptically symmetric distribution with location parameter µi =
(µi (a))a∈A and scale matrix Σi which is positive definite, there exists a mixed
strategy Nash equilibrium for all α ∈ (0.5, 1]n .
Proof. For each i ∈ I, BRiαi (τ−i ) is a convex set for all αi ∈ (0.5, 1] because
i
the payoff function uα
i (·, τ−i ) defined by (3.10) is a concave function of τi from
Lemma 3.5. From (3.10), uiαi (·) is a continuous function of τ . Then, the proof
follows from the similar arguments used in Theorem 3.2.
Remark 3.8. For each i ∈ I, if the random payoff vector (riw (a))a∈A has
strictly positive density then Theorem 3.7 holds for all α ∈ [0.5, 1]n .
Acknowledgements
This research was supported by Fondation DIGITEO, SUN grant No. 20140822D.
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